F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.73182.1 Research Article Articles Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)

[version 1; peer review: 1 approved with reservations, 1 not approved]

 Thangaraj Shruthi Conceptualization Data Curation Formal Analysis Investigation Methodology Software Writing – Original Draft Preparation 1 Goh Vik Tor Methodology Supervision Writing – Review & Editing https://orcid.org/0000-0003-4573-9493 a 1 Yap Timothy Tzen Vun Methodology Supervision Writing – Review & Editing https://orcid.org/0000-0003-1277-1710 2 Faculty of Engineering, Multimedia University, Cyberjaya, Selangor, 63100, Malaysia Faculty of Computing and Informatics, Multimedia University, Cyberjaya, Selangor, 63100, Malaysia vtgoh@mmu.edu.my

No competing interests were disclosed.

 28 2 2022 2022 11 246 11 2 2022 Copyright: © 2022 Thangaraj S et al. 2022 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Background: Smart grid systems require high-quality Phasor Measurement Unit (PMU) data for proper operation, control, and decision-making. Missing PMU data may lead to improper actions or even blackouts. While the conventional cubic interpolation methods based on the solution of a set of linear equations to solve for the cubic spline coefficients have been applied by many researchers for interpolation of missing data, the computational complexity increases non-linearly with increasing data size.

Methods: In this work, a modified recurrent equation-based cubic spline interpolation procedure for recovering missing PMU data is proposed. The recurrent equation-based method makes the computations of spline constants simpler. Using PMU data from the State Load Despatch Center (SLDC) in Madhya Pradesh, India, a comparison of the root mean square error (RMSE) values and time of calculation (ToC) is calculated for both methods.

Results: The modified recurrent relation method could retrieve missing values 10 times faster when compared to the conventional cubic interpolation method based on the solution of a set of linear equations. The RMSE values have shown the proposed method is effective even for special cases of missing values (edges, continuous missing values).

Conclusions: The proposed method can retrieve any number of missing values at any location using observed data with a minimal number of calculations.

 phasor measurement unit missing data data recovery smart grid interpolation cubic spline data quality data pre-processing The author(s) declared that no grants were involved in supporting this work. Introduction

The worldwide growing power systems highlight the need for better monitoring and control mechanisms to avoid major blackouts. Smart grids are intelligent systems that facilitate the development of communication, network, and computing technologies, protocols, and standards to integrate power system elements for two-way communication. This time-synchronized high-precision measurement device that is also known as a synchrophasor or Phasor Measurement Unit (PMU), gives clear information on the working of the entire grid. The PMU is used to monitor and control the power grid. It can help in providing real-time measurements by eliminating adverse conditions like blackouts. These combined characteristics of data availability, timeliness, and communication network contribute to the better performance of the PMU system. Although the role, impact, 1 architecture, technology, 2 applications, functionality, standards, and evolution of PMU (timing, measurement, communication, and data storage) have been released since 1995, the North American Synchro Phasor Initiative (NASPI) has highlighted the importance of data quality. 3 Data quality issues, their potential causes, and consequences are elaborated. 4 – 6 Generally, incomplete or missing data might affect the functionality of the entire system. 7 Hence, a way to handle missing values in PMU is mandatory for the effective functioning of the entire grid system.

In this paper, a modified recurrent equation-based method termed the Alpha Method (AM) for PMU missing data problem is proposed. The results are compared with the tri-diagonal matrix-based conventional cubic spline interpolation for the spline coefficients which is also termed the Linear Equations Method (LEM).

 Literature review

The need to fill in the missing values in PMU and potential causes have been reviewed. 5 – 7 These works imply the need for missing data recovery techniques for PMU data to enhance the accuracy of the decision-making process and show the data quality and security risks associated with the missing data in PMU. One of the popular approaches is the matrix completion (MC) based on missing data recovery. 8 – 12 The MC is the most exploited technique, however, a few of these were only theoretical approaches and a few approaches were only tested with simulated data.

Interpolation-based missing data recovery techniques 13 – 15 propose a reconstruction of missing values by a spatial interpolation or spatio-temporal interpolation of the values. Yet they require historical data of the same channel’s or time’s data for the interpolation. A few of the advanced/hybrid approaches 16 , 17 like k-nearest-neighbor and recurrent relation-based interpolations are not yet applied over the PMU data.

Missing data is a common problem in all fields of study; hence a variety of solutions are found to be effective based on the data pattern, data processing model, and data quality needs. However, adopting any conventional techniques available for treating missing values can get complex especially when solving the high precision and volume of PMU data. 15 Therefore, there is a need for a missing data recovery method for PMU data. NASPI presents a variety of data requirements, attributes, and data quality problems for both static data and real-time data. There is a need for designing an effective data recovery method to work without the need for historical data processing and training. 3 So, a data-driven recovery technique capable of recovering missing entries with available or observed data is much needed. Moreover, the technique should not get complex and time-consuming when the size of the data grows.

 Methods

Cubic spline interpolation is a widely used polynomial interpolation method for functions of one variable. Let f be a function from R to R . It is assumed that the value of f is known only at x 1 ≤ x 2 . ≤ x i … ≤ x n and let   f x i = a i . Piecewise cubic spline interpolation is the problem of finding the b i , c i and d i coefficients of the cubic polynomials SF i for 0 ≤ i ≤ n − 1 written in the form: SF i x = a i + b i x − x i + c i x − x i 2 + d i x − x i 3

Where x can take any value between x i and x i + 1 . That is, SF i x i = a i

Let the first-order derivative of equation (1) be: SF i ′ x = b i + 2 c i x − x i + 3 d i x − x i 2

The first-order derivative at x i for values of 1 ≤ i ≤ n − 1 will be SF i ′ x i = b i

And the second-order derivative be: SF i ′ ′ x = 2 c i + 6 d i x − x i

The second-order derivative at x i for values of 1 ≤ i ≤ n − 1 will be: SF i ′ ′ x i = 2 c i

For a smooth fit between the adjacent pieces, the cubic spline interpolation requires that the following conditions hold:

The cubic functions should intersect at the points left and right, for i = 0 to n − 1 SF i x i + 1 = SF i + 1 x i = a i + 1

For each cubic function to join smoothly with its neighbors, the splines should have continuous first and second derivatives at the data points i = 1 , … , n − 1 : SF i ′ x i + 1 = SF i + 1 ′ x i = b i + 1 SF i ′ ′ x i + 1 = SF i + 1 ′ ′ x i = 2 . c i + 1

If h i = x i + 1 − x i and if h i is equal for all i values, following Revesz, 17 the relation between coefficients a i and c i can be resolved: c i − 1 + 4 c i + c i + 1 = 3 h 2 a i − 1 − 2 a i + a i + 1 b i = a i + 1 − a i 1 h i − 2 c i + c i + 1 3 h i d i = 1 3 . h i c i + 1 − c i

Equation (6) represents a system of linear equations for the unknowns c i for 0 ≤ i ≤ n . As the values of a i are known, the value of c i can be found by solving the tri-diagonal matrix-vector equation Ax = B . While there are n+1 numbers of c i constants, equation (6) yields only ( n-2) equations. Based on the nature or type of spline assumed two more equations representing the boundary conditions of the spline. In general, two types of splines may be considered: natural cubic spline and clamped cubic spline.

For natural cubic spline interpolation, the following boundary conditions are assumed: c 0 = c n = 0.0 . That is, the second derivatives of the splines at the endpoints are assumed to be zero. Based on equation (4), a system of ( N+1) linear equations of ( N+1) variables can be formulated as: A = 1 0 0 0 ⋯ 1 4 1 0 ⋯ 0 1 4 1 ⋯ 0 0 0 0 0 0 0 0 0 0 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 ⋯ 0 0 0 0 0 0 0 0 ⋯ ⋯ ⋮ ⋮ ⋮ ⋮ 1 4 1 0 0 0 1 0 4 0 1 1 , x = c 0 c 1 ⋮ c n , and B = 0 3 h 1 a 0 − 2 a 1 + a 2 ⋮ 3 h 1 a n − 2 − 2 a n − 1 + a n 0

For clamped cubic spline interpolation the following boundary conditions are assumed: b 0 = f ′ ( x 0 ) and b n = f ′ ( x n ), where the derivatives f ′ ( x 0 ) and f ′ ( x n ), are known constants. Thus, based on the boundary conditions assumed both natural and cubic splines result in n+1 system of linear equations. The resulting system of n+1 linear equations can be used to get unique solutions by any of the standard methods for solving a system of linear equations.

Once the values of c i are obtained, using equations (5) and (6) respectively, the values of coefficients b i and d i can also be found. Similarly, under clamped spline interpolation, A = 2 1 0 0 ⋯ 1 4 1 0 ⋯ 0 1 4 1 ⋯ 0 0 0 0 0 0 0 0 0 0 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 ⋯ 0 0 0 0 0 0 0 0 ⋯ ⋯ ⋮ ⋮ ⋮ ⋮ 1 4 1 0 0 0 1 0 4 1 1 2 ,   x = c 0 c 1 ⋮ c n and B = 3 h 2 a 1 − a 0 − 3 h f ' x 0 3 h 1 a 0 − 2 a 1 + a 2 ⋮ 3 h 1 a n − 2 − 2 a n − 1 + a n 3 h f ' x 0 − 3 h 2 a n − a n − 1

Recurrence equation-based solution

Revesz, 17 chose boundary conditions that need to solve the tri-diagonal system given in equation (6) where x i rational variables e i rational constants, r is a non-zero rational constant and A is: A = r 1 0 0 ⋯ 1 4 1 0 ⋯ 0 1 4 1 ⋯ 0 0 0 0 0 0 0 0 0 0 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 ⋯ 0 0 0 0 0 0 0 0 ⋯ ⋯ ⋮ ⋮ ⋮ ⋮ 1 4 1 0 0 0 1 0 4 0 1 1 , x = x 1 x 2 ⋮ x n − 1 x n and b = e 1 e 2 ⋮ e n − 1 e n

The first row of the new matrix in (6) is shown to be equivalent to the first row of the clamped b matrix e 1 is e 1 = 3 r 2 h a 1 − a 0 h − f ′ x 0 + 1 r 2 c ~ 1 where, c ~ 1 is an estimate of c 1 and r = 2+ √ 3 ≈ 3.732 .

The chosen boundary conditions are such that the first row of the new matrix was the same as that of clamped cubic spline and while that of the last row was that of the natural cubic spline fixing the value of c n as 0. x i + x i + 1 r = ∑ 0 ≤ k ≤ i − 1 − 1 k e i − k r k + 1

Let ∝ 0 , ∝ i for 1 < i ≤ n − 1 and ∝ n , respectively be: ∝ 0 = 0 ∝ i = e i − ∝ i − 1 r = ∑ 0 ≤ k ≤ i − 1 − 1 k e i − k r k + 1 ∝ n = e n

Based on the above, the closed form of solution for x i can be given as: for x i = ∑ 0 ≤ k ≤ n − i − 1 r − k ∝ i + k

The above equation solves x i no matter exactly what the initial values for e i . This leads to a faster evaluation of the cubic spline than solving a tri-diagonal system. The major advantage of the method is when new measurements are added to the system. While conventional tri-diagonal matrix-based algorithm requires a complete redo of the entire computation, equation (14) leads to a faster update for each i ≤ n only with the addition of the term: − 1 r n + 1 − i ∝ n + i and x n + 1 = ∝ n + 1 . Similarly, ∝ i constants can be updated by adding a single term e n + 1

The system of linear equations given in equation (7), in general, is solved by the standard solution of linear equations in the matrix form Ax = b . Alternatively, it could be solved for n variables by the recurrence relations given equations (16) and (17). The two methods, the first using the tri-diagonal matrix-based solution for the spline coefficients is termed the Linear Equations Method (LEM) and the second one using recurrence relations is termed the Alpha Method (AM). The algorithmic procedure for LEM and AM are given below.

 Algorithmic procedure for regular tridiagonal matrix-based Linear Equation Method (LE)

Step 1: Given the initial vector with missing values, separate them into two sets of vectors, the observed values vector R obs and the missing values vector R Miss , having sizes of NO and NM, respectively, such that NO+ NM= N.

Step 2: R obs vector at x i values of the ( NO-1) splines shall be the a i coefficient vector.

Step 3: Using a i , generate the RHS vector E given in equation (11).

Step 4: Generate a square coefficient matrix A as given in equation (11)

Step 5: Solve for the c i vector is given in (11), using the relation Ac= E

Step 6: Applying c i in equations compute the b i and d i coefficient vectors for n-2 points of the R obs ,

Step 7: Using the values of a i , b i , c i and d i , missing values can be found by the equation (1) re-written as: SF i x = a i − b i x i + c i x i 2 − d i x i 3 + b i − 2 c i x i + 3 d i x i 2 x + c i − 3 d i x i x 2

Where x represents the missing positions, between x i and x i + 1 of spline i.

Algorithmic procedure for recurrent equation-based Alpha Method (AM)

Step 1: Given the initial vector with missing values, separate them into two sets of vectors, the observed values vector R obs and the missing values vector R Miss , having sizes of NO and NM, respectively, such that NO+ NM= N.

Step 2: The R obs vector at x i values of the ( NO-1) splines are the a i coefficient vector.

Step 3: Using a i , generate the RHS vector E given in equation (11).

Step 4: Set ∝ 0 = 0 and ∝ n = e n , given in equation (11) calculate the alpha vector using the relation.

∝ i = e i − ∝ i − 1 r = ∑ 0 ≤ k ≤ i − 1 − 1 k e i − k r k + 1 for i values ranging from 1 to NO-1

Step 5: Set x n = ∝ n and solve for c i values using the relation. c i = ∑ 0 ≤ k ≤ n − i − 1 r k ∝ i + k

Step 6: Applying c i in equations compute the b i and d i coefficient vectors for n-2 points of the R obs ,

Step 7: Using the values of a i , b i , c i and d i , missing values can also be found using equation (18), re-written here again for convenience: SF i x = a i − b i x i + c i x i 2 − d i x i 3 + b i − 2 c i x i + 3 d i x i 2 x + c i − 3 d i x i x 2

Where x represents the missing positions, between x i and x i + 1 of spline i.

The modifications are as follows: In the AM method rather than computing E, alpha vectors and c i coefficients for the full range of NO-1 data points only the RHS, E vector, was calculated for the full range of NO-1 data points, while alpha vector and c i were calculated only for i and i + 1 data elements, where i is the missing data element. For the imputation of i the element, only the E i vector for all NO-1 data points, ∝ i vector and c i vectors for i and i + 1 and b i and d i coefficients were essential for the calculation i th missing element and its imputation.

In addition, using the AM, an effective procedure was demonstrated for the computation of the following cases: (i) missing first and the last element of the data vector, (ii) missing multiple data points at the beginning and the end, and (iii) missing multiple elements anywhere in the data vector. That is in equation (18), when the current values of A [ i] are replaced either with A [ N-1] or A [ i-1] based on the position of missing edge values or continuous values the ToC and RMSE values have improved significantly.

 Results and discussion

A comparison between LEM and AM methods is shown here for the imputation of one-min real PMU system data having a size of 1490 data points for each of the 25 heterogeneous variables obtained from five different PMUs. Since our data does not have any missing values we artificially introduced the missing values of 10%, 20%, 30% in random.

A sample of one minute PMU data for five PMUs’ was used in the study. 18 One minute of PMU data with 10%, 20%, 30% missing data respectively for five PMUs were evaluated.

When the AM method was employed, the average root mean squared error (RMSE) values were 0.5968, 0.9448, and 1.2445 for 10%, 20%, and 30% of missing PMU data respectively. This can be seen in Figure 1. Moreover, for the same performance, the AM method showed significant improvements in its time of calculation (ToC) as shown in Figure 2. The average ToCs for the proposed AM method were 2.132, 1.9634, and 1.738s when recovering 10%, 20%, and 30% of its missing data. By comparison, LEM had ToC values of 32.7679, 33.4482, and 36.7988s for 10%, 20%, and 30% of its missing data, respectively. The proposed method reduced the ToC by a factor of approximately 10 times.

Comparison of RMSE values: (A) alpha method; (B) linear equation method. Comparison of Time of Calculation (ToC): (A) alpha method; (B) linear equation method. Conclusions

In this study, the proposed AM method was compared with the LEM technique. However, because of the proliferation of the data, there is a need for customization of this technique to handle a high volume of data to reduce computational time and power. In the proposed method, the approaches demonstrated a reduced computational effort and time of calculation for solving the coefficient vectors. This study has made the following contributions: (i) the recurrent relation-based alpha method has been effectively employed in the imputation of PMU data and its advantages are demonstrated as an effective and efficient alternative to the conventional technique, and (ii) an effective procedure for handling special cases (edge, continuous values) is shown, which has not been addressed clearly in other methods. The proposed method has proven effective, and it only requires 10% effort in comparison to the LEM. Future research will focus on the application of the modified recurrent method in the analysis of real-time or stream PMU data.

 Data availability Underlying data

Harvard Dataverse: Underlying data for ‘Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)’, ‘PMU data’, https://doi.org/10.7910/DVN/Y2LLJJ. 18

This project contains the following underlying data:

Data file: pmu1-1m-10.tab – One minute of data from PMU1 with 10% missing data

Data file: pmu1-1m-20.tab – One minute of data from PMU1 with 20% missing data

Data file: pmu1-1m-30.tab – One minute of data from PMU1 with 30% missing data

Data file: pmu2-1m-10.tab – One minute of data from PMU2 with 10% missing data

Data file: pmu2-1m-20.tab – One minute of data from PMU2 with 20% missing data

Data file: pmu2-1m-30.tab – One minute of data from PMU2 with 30% missing data

Data file: pmu3-1m-10.tab – One minute of data from PMU3 with 10% missing data

Data file: pmu3-1m-20.tab – One minute of data from PMU3 with 20% missing data

Data file: pmu3-1m-30.tab – One minute of data from PMU3 with 30% missing data

Data file: pmu4-1m-10.tab – One minute of data from PMU4 with 10% missing data

Data file: pmu4-1m-20.tab – One minute of data from PMU4 with 20% missing data

Data file: pmu4-1m-30.tab – One minute of data from PMU4 with 30% missing data

Data file: pmu5-1m-10.tab – One minute of data from PMU5 with 10% missing data

Data file: pmu5-1m-20.tab – One minute of data from PMU5 with 20% missing data

Data file: pmu5-1m-30.tab – One minute of data from PMU5 with 30% missing data

README.txt

Data are available under the terms of the Creative Commons Zero “No rights reserved” data waiver (CC0 1.0 Public domain dedication).

 References Phadke AG Bi T : Phasor measurement units, WAMS, and their applications in protection and control of power systems. J. Mod. Power Syst. Clean Energy. 2018;6(4):619–629. 10.1007/s40565-018-0423-3 Usman MU Faruque MO : Applications of synchrophasor technologies in power systems. J. Mod. Power Syst. Clean Energy. 2019;7(2):211–226. 10.1007/s40565-018-0455-8 Amidan B : Data Mining Techniques and Tools for Synchrophasor Data. North American SynchroPhasor Initiative (NASPI);2019, January;45. Miller LE : PMU Data Quality: A Framework for the Attributes of PMU Data Quality and a Methodology for Examining Data Quality Impacts to Synchrophasor Applications. 2017; no.March: pp.1–77. Huang C : Data quality issues for synchrophasor applications Part I: a review. J. Mod. Power Syst. Clean Energy. 2016;4(3):342–352. 10.1007/s40565-016-0217-4 Huang C : Data quality issues for synchrophasor applications Part II: problem formulation and potential solutions. J. Mod. Power Syst. Clean Energy. 2016;4(3):353–361. 10.1007/s40565-016-0213-8 Fang X : PMU Data Quality: A Framework for the Attributes of PMU Data Quality and a Methodology for Examining Data Quality Impacts to Synchrophasor Applications. IEEE Trans. Power Syst. 2017;7(1):1–6. Genes C Esnaola I Perlaza SM : Recovering missing Data via matrix completion in electricity distribution systems. IEEE Workshop on Signal Processing Advances in Wireless Communications, SPAWC, 2016-Augus (July 2016). 2016. Gao P Wang M Ghiocel SG : Missing Data Recovery by Exploiting Low-Dimensionality in Power System Synchrophasor Measurements. IEEE Trans. Power Syst. 2016;31(2):1006–1013. 10.1109/TPWRS.2015.2413935 Cai JF Candès EJ Shen Z : A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 2010;20(4):1956–1982. 10.1137/080738970 Genes C Esnaola II Perlaza SM : Recovering missing data via matrix completion in electricity distribution systems. IEEE Workshop on Signal Processing Advances in Wireless Communications, SPAWC, 2016-August, 1–6. 2016. Hastie T Mazu Missing Dataer R Lee JD : Matrix completion and low-rank SVD via fast alternating least squares. J. Mach. Learn. Res. 2015;16:3367–3402. 31130828 Gräler B Pebesma E Heuvelink G : Spatio-temporal interpolation using gstat. R Journal. 2016;8(1):204–218. 10.32614/RJ-2016-014 Cheng S Lu F : A two-step method for missing spatio-temporal Data reconstruction. ISPRS Int. J. Geo Inf. 2017;6(7). 10.3390/ijgi6070187 Deng M Fan Z Liu Q : A Hybrid Method for Interpolating Missing Data in Heterogeneous Spatio-Temporal Datasets. ISPRS Int. J. Geo Inf. 2016;5(2). 10.3390/ijgi5020013 Yang Z Liu H Bi T : A PMU data recovering method based on preferred selection strategy. Glob. Energy Interconnect. 2018;1(1):63–69. Revesz PZ : A recurrence equation-based solution for the cubic spline interpolation problem. International Journal of Mathematical Models and Methods in Applied Sciences. 2015;9(16):446–452. Thangaraj S Goh VT Yap TTV : PMU Data. 2021. Harvard Dataverse. 10.7910/DVN/Y2LLJJ 10.5256/f1000research.76818.r139819 Reviewer response for version 1 Farooq Shaik Mullapathi 1 Referee https://orcid.org/0000-0003-3630-8974 Department of Computer Science and Engineering, K. S. R. M. College of Engineering (UGC-Autonomous), Kadapa, Andhra Pradesh, India

Competing interests: No competing interests were disclosed.

 12 7 2022 Copyright: © 2022 Farooq SM 2022 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation reject

The manuscript proposes recurrent relation based alpha method to interpolate missing PMU data. Further, the authors try to prove that the proposed method reduces computational complexity. 

However, the comments are as follows, 

The Implementation of the proposed method is clearly missing (Hardware or software details used) in the manuscript which does not assist reproducing the results. 

Most of the manuscript is dedicated for theoretical discussion about the proposed method. But a comparison between the existing methods with the proposed method is missing. 

Add nomenclature that improves the readability of the manuscript. 

Only data set of PMU values are presented (PMU Data Harvard Data verse) instead need to add discussion about the details of PMU Data. 

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.

Are all the source data underlying the results available to ensure full reproducibility?

Partly

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

No

Reviewer Expertise:

Cyber security in smart grid communication network and VANET.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

Goh Vik Tor Multimedia University, Malaysia

Competing interests: No competing interests were disclosed.

 16 12 2022

The purpose of this preliminary paper is to introduce our work in missing data recovery using cubic spline interpolation, namely the mathematical foundation and algorithmic logic. These details have been presented and explained accordingly in the paper. Additionally, the data used is also available for download by interested parties. We aim to publish a more detailed paper soon which will contain more information such as those suggested by the reviewer. Thank you for the suggestion.

As stated earlier, the purpose of this preliminary paper is to introduce our proposed method, hence the emphasis on theoretical discussion. However, we have made an initial comparison with an existing method, namely the Linear Equation Method. This can be seen in the Results and discussion section.

The nomenclature is improved wherever possible to improve the readability of the manuscript.

The dataset presented in the work was obtained from a regional Electricity Authority in India. It was obtained for use as realistic data and brief details of the PMU data is now included. However, additional information such as the data source, the acquisition process, and the physical significance of the systemic variables are not detailed at this stage of algorithmic development as the main idea is only to demonstrate the efficacy of the missing data imputation algorithm. Nonetheless, we take note of this suggestion for our next submission. Thank you.

 10.5256/f1000research.76818.r125681 Reviewer response for version 1 Foo Mathias 1 Referee https://orcid.org/0000-0003-1400-2659 School of Engineering, University of Warwick, Coventry, UK

Competing interests: No competing interests were disclosed.

 10 3 2022 Copyright: © 2022 Foo M 2022 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve-with-reservations

In general, there is promising aspect of the proposed method but it has to be conveyed in a clearer manner. Here are my comments.

In Introduction section, the authors state that the comparison will be made with LEM. Can the author explain why specifically LEM is compared? Is that the current state-of-the-art method?

In Literature Review section, NASPI is mentioned but without proper definition of the acronym.

If my understanding is right, Equation (10) is a systems of linear equation of (7). Then, why does the h value in the B matrix have an exponent of 1 instead of 2 as of Equation (7)?

Statement above Equation (11): Unless I'm mistaken, there is no d coefficient to be solved from either equations (5) or (6).

Equation (11): Like Equation (10), can the authors clarify why the exponent of 1 is used for h?

Equation (13): Why is r taking this value? A bit more explanation would be helpful.

Equation (16): Why is there a 'for' in the equation?

Step 3 of LE method: There is no vector E in Equation (11).

Step 3 of AM method: There is no vector E in Equation (11).

Step 4 of AM method: There is no alpha term in Equation (11).

Results and Discussions section: Can the author explicitly write down the equation for RMSE?

Also, I am quite surprised with the huge difference in terms of RMSE between the two methods even for the case of 10% missing data considering the same equation (18) is used for both algorithms. The difference in ToC is understandable, but the vast difference in RMSE is a bit out of my expectation. Could the author briefly comment on the plausible reason for this huge difference in the RMSE value despite both algorithm using equation (18).

Overall comment: The mathematical derivation is not easy to follow and there are potential mistakes in citing the equations, which makes it even harder to follow. Thus, it is difficult to ascertain whether the results can be reproduced. 

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Partly

Are all the source data underlying the results available to ensure full reproducibility?

Partly

Is the study design appropriate and is the work technically sound?

Partly

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

No

Reviewer Expertise:

Dynamical system modelling

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Goh Vik Tor Multimedia University, Malaysia

Competing interests: No competing interests were disclosed.

 16 12 2022

The idea of cubic spline is the development of a series of unique cubic polynomials that are fitted between the data points. Based on four continuity relations between points in the spline, the relationships between the spline coefficients shall result in a system of unique n x n linear equations in the matrix form Ax = B. The solution of this unique system of linear equations results in the values of constants at each spline point. Whenever some changes occur in any one of the splines, the system of linear equations must be solved for every specific change to fit the spline. The tri-diagonal method of solving a system of linear equations was employed in this study for comparison. Hence and from the reference, we made such a comparison to linear equation method.

The word is already introduced in the introduction section as North American Synchro Phasor Initiative (NASPI).

The exponent for h in Equation (10) and (11) should be 2 instead of 1. We have made the corrections accordingly. Thank you.

As the reviewer correctly noted, the d coefficient is not solved in Equations (5) or (6). It is instead solved using Equation (9). We have made the amendments accordingly.

The r-value in Equation (13) is a non-zero rational constant; the value is adapted from our reference work [17]. We have added this citation in the text.

The ‘ for’ should not be in Equation (16). It has been corrected.

Step 3 and Step 5 of the LE method has been corrected as vector B instead of E.

Reference to Equation (11) in Step 4 of the AM method has been removed.

The equation for RMSE has been added in the text.

Upon inspection, we found that the variables b, c, and d of the cubic spline is found to be similar in both methods as the reviewer correctly predicted. We re-examined our results and have determined that due to an oversight, an error occurred in the final calculations of imputation values in the LE method. Instead of the coordinate numbers, the spline values at the coordinates were used for the calculation of missing values. As such, we have made the corrections to the results and discussions, as well as the plots in Figure 1 and Figure 2.

All corrections and suggestions given were incorporated and the revised version of the paper is written.  We really appreciate the time spent by the reviewer for the useful suggestions and corrections made.